# Finite Element Analysis of Fixed Orthodontic Retainers

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{2}), nickel titanium, or cobalt-chromium. The material selection directly influences the RS due to the resulting differences in Young’s modulus and might therefore affect the transmission of force from a loaded tooth to the neighbouring teeth.

## 2. Materials and Methods

_{2}(Zahnwerkstatt Wernigerode, Germany).

#### 2.1. Bending Stiffness of Multistranded Retainers

#### 2.2. Finite Element Model

#### 2.2.1. Tooth Resilience

#### 2.2.2. Retainer Design, Attachment, and Loading Conditions

_{0}) to give the normalised bending stiffness (k/k

_{0}). The list of the simulated retainers, ordered by increasing bending stiffness, is given in Table 4.

_{0}) will be used to refer to the individual retainers throughout this investigation. Additional FE computations were carried out with the largest RD and an extremely high Young’s modulus (enlarged by 10

^{6}) to simulate the effect of a rigid retainer.

#### 2.3. Analysed Data

_{trans}/F

_{res}) and RS (k) using an arbitrary reference stiffness (k

_{0}) (least stiff retainer) as well as parameters a and b was suited to fit the FE computation data as it shows asymptotic behaviour towards an upper (F

_{rel,rigid}, relative force transmission for a rigid retainer) and lower threshold (no force transmission if the retainer has no stiffness).

_{u}= 20 MPa) for the interface between the teeth and the adhesive [20,21,22,23]. The resulting value represents the utilised capacity of the adhesive bond for the corresponding combination of the RS, TR, and load case. A high degree of utilised capacity in a certain area therefore corresponds to high normal and resulting shear stresses in that area.

## 3. Results

#### 3.1. Experimental Testing

#### 3.2. Force Transmission to the Bonding Area

_{0}= 1). The force transmission values within this group were the highest for the 0.081 mm, six-strand twisted stainless steel retainer (no. 2; about 27% and 36% for LC1 and LC2, respectively). The force transmission values for all the other conventional retainers were in between. The ZrO

_{2}CAD/CAM retainers (no. 6) were included as an extreme example of retainers with a high RS and showed force transmission values up to 4.8 times higher than those for the stiffest conventional retainer (at low TR). Applying an oblique bite force increased the force transmission, making this load case more critical for debonding.

^{6}increased the relative force transmission to the upper threshold, corresponding to a complete rigid retainer, which amounted to F

_{rel,rigid}= 79.4% for LC1 and F

_{rel,rigid}= 90.1% for LC2 (see Formula (3)). The fitted parameters a and b for the exponential function given in Formula (3) are listed in Table 6.

#### 3.3. Effect of Retainer Diameter on Stress Distribution

_{0}= 1 and k/k

_{0}= 12.2), the increase in RS (here by a factor of 12.2) increased the utilised capacity. Although this was the case for both the increase in RD (dashed blue curve in Figure 4) and a switch to the solid cross section (solid blue curve), the maximum degree of utilised capacity increased less with the increase in RD. This effect of a more uniform stress distribution and therefore the distribution of the utilised capacity with a thicker retainer was more prominent with a higher RS. Comparing the dashed blue line with the dashed red line, the RS and accordingly the force transmission increased only by a rising RD. However, the maximum utilised capacity at tooth 31 decreased because the stress distribution was more uniform. This effect can be seen in Figure 5c. In contrast, by switching to a solid cross section (solid red curve) instead of changing the RD, the maximum utilised capacity increased slightly. Because the force transmission via the adhesive bond was nearly identical for both retainers, it was clear that the less-uniform stress distribution for the thinner retainer caused this effect. A good example of such a non-uniform stress distribution can be found in the solid blue curve in Figure 4 and in Figure 5b.

## 4. Discussion

_{2}and have various designs that have never been used to produce hand-bent fixed retainers before. The ZrO

_{2}retainer was included as an example of a fixed retainer with a particularly high RS, and the relative force transmission of this retainer was up to 4.8 times higher than that of the stiffest conventional fixed retainer. From a clinical point of view, such higher stiffness values, which correspond to higher force transmission and higher overall stress values (as long as the adhesive bonding area remains the same), might lead to a higher risk of adhesive failure. This might be relevant because these fixed retainers are already commercially available and being used on patients.

## 5. Conclusions

- Higher bending stiffness, higher tooth resilience, and an oblique load situation all increase force transmission and therefore overall stresses on the bonding area of the loaded tooth and its neighbour teeth.
- A lower retainer diameter decreases the uniformity of the stress distribution leading to local stress peaks within a small field of the bonding area.
- In conclusion, using fixed conventional or CAD/CAM retainers with rather low stiffness (up to 3 N/mm) might be recommended, especially in cases with high tooth resilience. Therefore, typical stainless steel multistranded retainers should not exceed a diameter of up to 0.55 mm. However, it is important to note that clinical studies have to validate the results of this FE study as accurate recommendations cannot be made based on a numerical simulation alone.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Booth, F.A.; Edelman, J.M.; Proffit, W.R. Twenty-year follow-up of patients with permanently bonded mandibular canine-to-canine retainers. Am. J. Orthod. Dentofac. Orthop.
**2008**, 133, 70–76. [Google Scholar] [CrossRef] [PubMed] - Chinvipas, N.; Hasegawa, Y.; Terada, K. Repeated bonding of fixed retainer increases the risk of enamel fracture. Odontology
**2014**, 102, 89–97. [Google Scholar] [CrossRef] [PubMed] - Aye, S.T.; Liu, S.; Byrne, E.; El-Angbawi, A. The prevalence of the failure of fixed orthodontic bonded retainers: A systematic review and meta-analysis. Eur. J. Orthod.
**2023**, 45, 645–661. [Google Scholar] [CrossRef] [PubMed] - Roser, C.J.; Bauer, C.; Hodecker, L.; Zenthofer, A.; Lux, C.J.; Rues, S. Comparison of six different CAD/CAM retainers vs. the stainless steel twistflex retainer: An in vitro investigation of survival rate and stability. J. Orofac. Orthop.
**2023**. [Google Scholar] [CrossRef] - Almeida, E.O.; Rocha, E.P.; Freitas Junior, A.C.; Anchieta, R.B.; Poveda, R.; Gupta, N.; Coelho, P.G. Tilted and short implants supporting fixed prosthesis in an atrophic maxilla: A 3D-FEA biomechanical evaluation. Clin. Implant. Dent. Relat. Res.
**2015**, 17 (Suppl. S1), e332–e342. [Google Scholar] [CrossRef] [PubMed] - Lisiak-Myszke, M.; Marciniak, D.; Bielinski, M.; Sobczak, H.; Garbacewicz, L.; Drogoszewska, B. Application of Finite Element Analysis in Oral and Maxillofacial Surgery-A Literature Review. Materials
**2020**, 13, 3063. [Google Scholar] [CrossRef] [PubMed] - Conserva, E.; Consolo, U.; Sancho, A.G.; Foschi, F.; Paolone, G.; Giovarrusscio, M.; Sauro, S. Stress distribution in carbon-post applied with different composite core materials: A three-dimensional finite element analysis. J. Adhes. Sci. Technol.
**2017**, 31, 2435–2444. [Google Scholar] [CrossRef] - Tsumanuma, K.T.S.; Caldas, R.A.; Silva, I.D.; Miranda, M.E.; Brandt, W.C.; Vitti, R.P. Finite Element Analysis of Stress in Anterior Prosthetic Rehabilitation with Zirconia Implants with and without Cantilever. Eur. J. Dent.
**2021**, 15, 669–674. [Google Scholar] [CrossRef] [PubMed] - Gomes, E.A.; Diana, H.H.; Oliveira, J.S.; Silva-Sousa, Y.T.; Faria, A.C.; Ribeiro, R.F. Reliability of FEA on the Results of Mechanical Properties of Materials. Braz. Dent. J.
**2015**, 26, 667–670. [Google Scholar] [CrossRef] [PubMed] - Singh, J.R.; Kambalyal, P.; Jain, M.; Khandelwal, P. Revolution in Orthodontics: Finite element analysis. J. Int. Soc. Prev. Community Dent.
**2016**, 6, 110–114. [Google Scholar] [CrossRef] [PubMed] - Zachrisson, B.J. Third-generation mandibular bonded lingual 3-3 retainer. J. Clin. Orthod.
**1995**, 29, 39–48. [Google Scholar] - Zachrisson, B.U. Differential retention with bonded retainers. World J. Orthod.
**2007**, 8, 190–196. [Google Scholar] [PubMed] - Zachrisson, B.U. Multistranded wire bonded retainers: From start to success. Am. J. Orthod. Dentofac. Orthop.
**2015**, 148, 724–727. [Google Scholar] [CrossRef] [PubMed] - Milheiro, A.; de Jager, N.; Feilzer, A.J.; Kleverlaan, C.J. In vitro debonding of orthodontic retainers analyzed with finite element analysis. Eur. J. Orthod.
**2015**, 37, 491–496. [Google Scholar] [CrossRef] - Pornamazeh, T.; Geramy, A.; Heidari, S.; Rajabizadeh, M.; Kamali, E.; Ghadirian, H. Comparison of the debonding force of metal, glass and polyethylene Fiber reinforced composite retainers: Mechanical and finite element analyses. Int. Orthod.
**2022**, 20, 100685. [Google Scholar] [CrossRef] - Roser, C.J.; Rues, S.; Erber, R.; Hodecker, L.; Lux, C.J.; Bauer, C.A.J. Tooth mobility restriction by multistranded and CAD/CAM retainers-an in vitro study. Eur. J. Orthod.
**2024**, 46, cjad076. [Google Scholar] [CrossRef] - DIN EN ISO 15841; Dentistry—Wires for Use in Orthodontics (ISO 15841:2014). German Version EN ISO 15841:2014. Beuth Verlag: Berlin, Germany, 2014.
- Roser, C.J.; Zenthofer, A.; Lux, C.J.; Rues, S. A new CAD/CAM tooth mobility simulating model for dental in vitro investigations. Clin. Oral Investig.
**2023**, 27, 5131–5140. [Google Scholar] [CrossRef] - Boldt, J.; Knapp, W.; Proff, P.; Rottner, K.; Richter, E.J. Measurement of tooth and implant mobility under physiological loading conditions. Ann. Anat.
**2012**, 194, 185–189. [Google Scholar] [CrossRef] [PubMed] - Richter, C.; Jost-Brinkmann, P.G. Shear bond strength of different adhesives tested in accordance with DIN 13990-1/-2 and using various methods of enamel conditioning. J. Orofac. Orthop.
**2015**, 76, 175–187. [Google Scholar] [CrossRef] - Blocher, S.; Frankenberger, R.; Hellak, A.; Schauseil, M.; Roggendorf, M.J.; Korbmacher-Steiner, H.M. Effect on enamel shear bond strength of adding microsilver and nanosilver particles to the primer of an orthodontic adhesive. BMC Oral Health
**2015**, 15, 42. [Google Scholar] [CrossRef] - Roser, C.J.; Ruckschloss, T.; Zenthofer, A.; Rammelsberg, P.; Lux, C.J.; Rues, S. Orthodontic shear bond strength and ultimate load tests of CAD/CAM produced artificial teeth. Clin. Oral Investig.
**2022**, 26, 7149–7155. [Google Scholar] [CrossRef] - Ruttermann, S.; Braun, A.; Janda, R. Shear bond strength and fracture analysis of human vs. bovine teeth. PLoS ONE
**2013**, 8, e59181. [Google Scholar] [CrossRef] [PubMed] - Möhlhenrich, S.C.; Jäger, F.; Jäger, A.; Schumacher, P.; Wolf, M.; Fritz, U.; Bourauel, C. Biomechanical properties of CAD/CAM-individualized nickel-titanium lingual retainers: An in vitro study. J. Orofac. Orthop. Fortschr. Kieferorthopädie
**2018**, 79, 309–319. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Schematic illustration of the three-point bending test setup and (

**b**) close-up of the relevant parts with a triple-stranded stainless steel retainer during testing.

**Figure 2.**(

**a**) Cross section of tooth 31 with all its components including the two load cases and (

**b**) close-up of the model with the least and most stiff retainer.

**Figure 3.**Force transmission (F

_{trans}) to the adhesive bond of the loaded tooth divided by the applied force (F

_{res}) over bending stiffness (k), normalised to the least stiff retainer (k

_{0}, no. 4). Data points are fitted with a function of the type seen in Formula (3). The area of conventional hand-bent retainers is highlighted and enlarged with a green rectangle.

**Figure 4.**Maximum degree of utilised adhesive bond capacity plotted for each tooth for five different retainers. The retainers demonstrate the effects of a variation in the RD configuration. The least stiff retainer (k/k

_{0}= 1) corresponds to the solid green curve. An increase in the RD is displayed with the dashed blue curve while the switch to a solid cross section is displayed with the solid blue curve. A further increase in the RD is displayed with the dashed red curve while the configuration that is different to the dashed blue curve is displayed with the solid red curve. The solid blue and red curves also demonstrate the effect of an increase in the RD. All the displayed models correspond to high TR and LC1.

**Figure 5.**Degree of utilised capacity for three different retainers. The values were calculated according to the fracture hypothesis as described in Formula (4) (σ/σ

_{u}+ τ/τ

_{u}), where σ is the normal stress, σ

_{u}the tensile bond strength, τ the resulting shear stress, and τ

_{u}the shear bond strength. (

**a**) k/k

_{0}= 1 with RD = 0.38 mm in the multistranded configuration, (

**b**) k/k

_{0}= 12.2 with RD = 0.4 mm and a solid cross section, and (

**c**) k/k

_{0}= 100.1 with RD = 1.20 mm in the multistranded configuration. (

**a**) corresponds to the solid green curve in Figure 4, (

**b**) to the solid blue curve, and (

**c**) to the dashed red curve. All displayed models correspond to high TR and LC1.

**Figure 6.**Maximum degree of utilised adhesive bond capacity plotted for each tooth for the same retainer but a different load case (dashed line) and a higher TR (blue).

**Table 1.**Details of the retainers included in this study. Retainers with a wide range of diameters were included to cover the range of commercially available retainers.

Number | Retainer Name | Manufacturer | Material | Diameter RD | Configuration |
---|---|---|---|---|---|

1 | Dentaflex | Dentaurum, Ispringen, Germany | Stainless steel | 0.50 mm | 6-strand twisted, conventional (bendable) |

2 | Stainless steel lingual retainer | 3M Oralcare, Seefeld, Germany | Stainless steel | 0.81 mm | 3-strand twisted, conventional (bendable) |

3 | Penta Twist | Gold’n Braces, Tampa, USA | Stainless steel (gold coated) | 0.53 mm | 6-strand twisted, conventional (bendable) |

4 | Dentaflex | Dentaurum, Ispringen, Germany | Stainless steel | 0.38 mm | 6-strand twisted, conventional (bendable) |

5 | Titanium retainer wire | Dentaurum, Ispringen, Germany | Titanium grade 5 | 0.50 mm | 3-strand twisted, conventional (bendable) |

Level of TR | Vertical TR [µm/N] | Horizontal TR [µm/N] | Ratio [-] |
---|---|---|---|

Low | 1.05 | 4.22 | 4.02 |

Medium | 2.01 | 6.04 | 3.01 |

High | 2.99 | 9.33 | 3.12 |

Material | Young’s Modulus E [GPa] | Poisson’s Ratio ν [-] |
---|---|---|

Teeth (dentine) | 15 | 0.30 |

Adhesive (composite resin) | 6 | 0.30 |

Retainer | 13–200 | 0.30 |

Periodontal ligament | Chosen according to the desired TR |

**Table 4.**Combinations of simulated E and RD resulting in the relative retainer bending stiffness (ratio of retainer bending stiffness k to the bending stiffness of the least stiff retainer k

_{0}).

Young’s Modulus E [GPa] | Diameter RD [mm] | Relative Bending Stiffness k/k_{0} |
---|---|---|

20 * | 0.38 | 1 |

13 * | 0.53 | 2.5 |

200 | 0.34 | 6.5 |

200 | 0.40 | 12.2 |

20 * | 0.71 | 12.2 |

200 | 0.54 | 40.1 |

200 | 0.68 | 100.1 |

20 * | 1.20 | 100.1 |

200 | 0.80 | 194.5 |

200 | 1.20 | 984.7 |

Number | Stiffness k [N/mm] Mean Value (SD) | k/k_{0} [-] |
---|---|---|

1 | 2.97 (0.05) | 3.1 |

2 | 15.80 (0.43) | 16.6 |

3 | 2.36 (0.04) | 2.5 |

4 | 0.95 (0.02) | 1.0 |

5 | 1.76 (0.09) | 1.9 |

**Table 6.**Resulting parameters a and b of the fitting function (Formula (3)) associated to each level of TR and each LC. The stiffness of the most resilient retainer was taken as a reference stiffness k

_{0}.

Low TR | Medium TR | High TR | |
---|---|---|---|

LC1 | a = 4.55|b = 0.33 | a = 3.83|b = 0.38 | a = 3.34|b = 0.41 |

LC2 | a = 2.85|b = 0.27 | a = 2.51|b = 0.31 | a = 2.19|b = 0.32 |

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## Share and Cite

**MDPI and ACS Style**

Hetzler, S.; Rues, S.; Zenthöfer, A.; Rammelsberg, P.; Lux, C.J.; Roser, C.J.
Finite Element Analysis of Fixed Orthodontic Retainers. *Bioengineering* **2024**, *11*, 394.
https://doi.org/10.3390/bioengineering11040394

**AMA Style**

Hetzler S, Rues S, Zenthöfer A, Rammelsberg P, Lux CJ, Roser CJ.
Finite Element Analysis of Fixed Orthodontic Retainers. *Bioengineering*. 2024; 11(4):394.
https://doi.org/10.3390/bioengineering11040394

**Chicago/Turabian Style**

Hetzler, Sebastian, Stefan Rues, Andreas Zenthöfer, Peter Rammelsberg, Christopher J. Lux, and Christoph J. Roser.
2024. "Finite Element Analysis of Fixed Orthodontic Retainers" *Bioengineering* 11, no. 4: 394.
https://doi.org/10.3390/bioengineering11040394